Density Interpolation Research Articles

Gröbner bases and factorisation in discrete probability and Bayes

Grobner bases, elimination theory and factorization may be used to perform calculations in elementary discrete probability and more complex areas such as Bayesian networks (influence diagrams). The paper covers the application of computational algebraic geometry to probability theory. The application to the Boolean algebra of events is straightforward (and essentially known). The extension into the probability superstructure is via the polynomial interpolation of densities and log densities and this is used naturally in the Bayesian application.

Almost Free Rotation of NH3 Molecules in Crystals: Observation from a Maximum-Entropy Reconstruction of the Proton Density in [Co(NH3)6](PF6)2

Accurate density data yield the primary information required for a thermodynamic model of molecular disorder. A detailed analysis of the structure from neutron Bragg data shows that the protons of the ammonia groups perform nearly unhindered rotations at room temperature as well as at 30K. The nuclear density distribution is derived from a combination of conventional crystallographic split-atom density interpolation and maximum-entropy density reconstruction. The crystallographic density interpolation allows unique phases to be determined, i.e. phases that are independent of details of the model, for the measured Fourier components. MaxEnt reconstruction eliminates series-termination effects and provides for higher quality in the spatial resolution compared with standard Fourier maps. The proton density obtained in [Co(NH 3 ) 6 ](PF 6 ) 2 at both temperatures is concentrated on a plane perpendicular to the fourfold crystal axis and is found to be nearly circular, with a weak tetragonal contribution superimposed. This is a strong indication of nearly free uniaxial rotation in this compound. The proton density is analysed in terms of an anharmonic orientational potential, which couples rotational and translational motion. The nearly unhindered rotation in this compound is a consequence of the quasi-eightfold symmetry built up by the next-neighbour F atoms surrounding the NH 3 groups in their plane of rotation.

Application of Line-Length Related Interpolation Methods to Problems in Coal Preparation—I: Mathematical Basis and Simple Washability Data Interpolation

Hughes’ has proposed a method for dealing with the problems of interpolating coal washability data which eliminates many of the difficulties encountered in previously proposed procedures. In particular, the problem of the interpolating function becoming negative was solved, but the computational procedure proposed was not suited to convenient computation. Lyman modified the solution to provide a smoother interpolating function, recast the computations in a manner well-suited to fully-automatic operation, and generalised and further extended the method to deal with a more difficult problem relating to interpolation of bulk density, size distribution and density distributions as a function of bed depth in samples taken from a jig bed. These methods were implemented in a major project by the author. This paper presents the mathematical background to both applications and provides an example of their application to simple interpolation of washability information. The second paper of this pair provides examples of the more complex application of the method to data arising from sampling a jig bed in layers, and the third paper considers two-dimensional interpolation of washability data.

Current source density estimation and interpolation based on the spherical harmonic Fourier expansion.

A method for the spatial analysis of EEG and EP data, based on the spherical harmonic Fourier expansion (SHE) of scalp potential measurements, is described. This model provides efficient and accurate formulas for: (1) the computation of the surface Laplacian and (2) the interpolation of electrical potentials, current source densities, test statistics and other derived variables. Physiologically based simulation experiments show that the SHE method gives better estimates of the surface Laplacian than the commonly used finite difference method. Cross-validation studies for the objective comparison of different interpolation methods demonstrate the superiority of the SHE over the commonly used methods based on the weighted (inverse distance) average of the nearest three and four neighbor values.

The Exponential Accuracy of Fourier and Chebyshev Differencing Methods

It is shown that when differencing analytic functions using the pseudospectral Fourier or Chebyshev methods, the error committed decays to zero at an exponential rate.

Calculation of the phonon spectrum of potassium with a microscopic theory

The microscopic theory of force constants is employed for calculating the phonon frequencies of potassium. As in a preceeding paper we rely upon the translation invariance requirement in the form of a sum rule. Because of this sum rule and an interpolation of the electronic density we can avoid any approximation for the density-density correlation function. The resultant phonon spectrum is in good agreement with experiment.